**The External Shapes of periodic 2-D arrays of repeated two-dimensional motifs **

Point symmetry and lattice structure determine the possible shapes that two-dimensional crystals can assume. In order to investigate these shapes we draw the relevant 2-D lattice, choose a suitable axial system (i.e. a set of crystallographic axes) to which the orientation of the crystal faces can be referred, determine the possible Forms (a Form is a set of equivalent faces), and consider possible combinations of Forms (belonging to the same Crystal Class (Point Group)). These combinations (and also single, but closed Forms) represent the possible shapes of the Crystals.

**Oblique Net**

The symmetry of its typical oblique building block (considered as empty) determines the 2-D Oblique Crystal System. It has two Classes, **2** and **1**.

In Figure 1 a possible oblique net is drawn (notice that the drawing is __not__ meant to be a perspectivic one. All angles are real angles).

Figure 1. *A possible oblique net (2-D lattice). Two symmetry types of motifs can be periodically repeated according to this net. The possible symmetry types of the motifs are represented by the Point Groups ***2*** and ***1***. As indicated a set of crystallographic axes can be based on this net. See also Figure 2.*

Figure 2. *A chosen oblique axial system to describe faces and Forms.
(Of course in 2-D crystals the 'faces' are in fact just lines, but nevertheles we keep calling them faces). The *

Let us start with Point Group (= Crystal Class)

This means that the 2-D crystal to be constructed has, as its only point symmetry element, a 2-fold rotation axis. The only Plane Group belonging to this Class is

As we saw in our consideration about

Figure 3. *Some possible faces that cam be supported by the oblique net.*

We insert a possible initial face -- in this case a face parallel to one of the crystallographic axes. See the next Figure.

Figure 4. *In order to construct one of the possible Forms of the Class ***2*** we introduce a face. This face will be multiplied by the 2-fold rotation axis perpendicular to the plane of the drawing (indicated by a small solid blue ellipse) such that the resulting face configuration (Form) will have point symmetry ***2***. The next Figure illustrates the action of the 2-fold rotation axis resulting in an open Form consisting of two parallel faces.*

Figure 5. *When a face is introduced, such that it is parallel to one of the crystallographic axes, a second face, parallel to the initial one is implied. The result is an open Form, possessing indeed the point symmetry of the present Class *(**2**)*. As such it cannot (because it is open) constitute a *(2-D)* crystal. Only when it combines with other Forms of the same Crystal Class, such that a closed face configuration is the result, can it go into the constitution of a crystal.*

So we've for the first time constructed a

Another special Form can be derived from an initial face parallel to the other crystallographic axis. See Figures 6 and 7.

Figure 6. *Introduction of a face, parallel to the other of the crystallographic axes.*

Figure 7. *The presence of an initial face, as indicated in Figure 6, implies a second face parallel to the intial face, because of the action of the 2-fold rotation axis. The resulting face pair is again a special Form.*

We can also introduce a face with a more general orientation with respect to the crystallographic axes (Recall that these axes are not equivalent). See Figure 8.

Figure 8. *Introduction of a face with a more general orientation to the crystallographic axes implies -- because of the action of the 2-fold rotation axis -- a second face parallel to the initial one. The resulting face pair is an open, general Form.*

Other more general orientations of introduced faces are possible. The next Figure depicts one such face, together with a second face implied by it.

Figure 9. *Introduction of a face with a more general orientation with respect to the crystallographic axes implies a second face parallel to the initial one. The resulting face pair is yet another open, general Form.*

These Forms (each of which is a set of equivalent faces) can (and in the present case should) enter in

Figure 10. *A combination of the Forms of Figures 7 and 9. Because this combination is closed it can represent a *(2-D)* crystal. The red lines signify the two crystallographic axes, while the small blue solid ellipse signifies a 2-fold rotation axis, as the only point symmetry element of the crystal.
Of course the building blocks are imagined to be utterly small, resulting in the faces always to be smooth macroscopically.*

The next Figure shows the same 2-D crystal, without indication of crystallographic axes and the 2-fold rotation axis. The crystal shape emerges from a combination of two Forms

Figure 11. *Two-dimensional crystal.
Heavy solid blue lines indicate faces.*

Figure 12 shows the same crystal without the auxiliary lines indicating faces. The

Figure 12. *The two-dimensional crystal, drawn without any auxiliary features.
The faces *

Figures 11 and 12 show that if the faces

The next two Figures give two more possible Forms.

Figure 13. *A possible Form of the 2-D Crystal Class ***2*** of the 2-D Oblique Crystal System. The axial system, and the 2-fold rotation axis representing the point symmetry of the Form, are indicated.*

Figure 14. *A possible Form of the 2-D Crystal Class ***2*** of the 2-D Oblique Crystal System. The axial system, and the 2-fold rotation axis representing the point symmetry of the Form, are indicated.*

Figure 15. *Construction of a 2-D crystal from a combination of four Forms :
First Form : The two faces *

Second Form

Third Form

Fourth Form

The axial system, and the 2-fold rotation axis representing the point symmetry of the crystal, are indicated. The final result of the construction is given in the next Figure.

Figure 16. *A two-dimensional crystal constructed from the four Forms mentioned above.*

The final habit of the crystal, i.e. its intrinsic shape, is determined by several factors

- The nature of the crystal lattice, which determines the shape of the (empty) building block.
- The chemical composition, i.e. the nature of the motifs placed in the lattice.

The growth rate of a face is inversely proportional to the number of nodes it encounters.

Further we can say that in general the slowest growing faces (Forms) become prominent in the final shape of the crystal, because the fastest growing faces grow themselves rapidly out of existence, as the next Figure shows.

Figure 17. *The face ***b*** is the fastest growing face, but just because of that it grows itself out of existence.*

The next Figure depicts yet another possible combination of Forms

Figure 18. *A 2-D crystal of the Class ***2*** from a combination of two Forms.
The faces *

This implies that the crystal becomes elongate in the direction along the faces

Different faces of a crystal, grown from a solution or melt of a certain chemical substance, often show different

In our two-dimensional crystals we can show how we could imagine that different faces (i.e. non-equivalent faces) can show different

Figure 19. *The faces ***d*** and ***h*** of Figure 18, with the building blocks furnished with motif units. The motifs associated with the lattice nodes -- recall that these nodes are just theoretical constructs to indicate the periodic repetition of the motifs -- have a point symmetry *

If we consider the visible part of the structure of Figure 19 as a whole crystal, then we have four faces. Every two such faces that are parallel to each other are equivalent, they present the same atomic aspect to the growing environment of the crystal. See Figure 19a.

Figure 19a. *This 2-D crystal has four faces, ***d, d, h, h***. The faces *

Figure 20. *The face ( i ), seen in Figure 8, is here again depicted, with the building blocks furnished with motif units. The lattice is still according to the oblique net, and the motifs associated with the nodes again have point symmetry ***2***. But also here the motifs are incomplete at the face of the crystal.*

Figure 21. *The face ***e*** of Figure 11, with building blocks furnished with motif units and as such representing Plane Group ***P2***. Also here the motifs are incomplete at the crystal's surface.*

Figure 22. *The face ***g*** of Figure 14 and 15, with building blocks furnished with motif units and as such representing Plane Group ***P2***. Also here the motifs are incomplete at the crystal's surface, which is in the present case only evident when we consider the motif s.l.*

The Figures 19, 20, 21 and 22 clearly show that different faces can display

In the next Part we will discuss the 2-D Crystal Class

To continue, click HERE for Part Nine.